哈佛大学概率与数理统计课件

-HowtoBeAWinner-TheMathsofRaceFixingandMoneyLaundering

JohnDBarrowWhyisProbabilityTheoryNotAncient?ReligiousbeliefsOrNoconceptofequallylikelyoutcomes???“Andtheysaideveryonetohisfellow,Come,andletuscastlots,thatwemayknowforwhosecausethisevilisuponus.SotheycastlotsandthelotfelluponJonah.”BookofJonah1v7St.Augustine:“Wesaythatthosecausesthataresaidtobebychancearenotnonexistentbutarehidden,andweattributethemtothewillofthetrueGod”Sheep's’anklebones,6-sided,numbered,asymmetricalDivinationwithsetsof5inAsiaMinorfrom3600BCEventuallyreplacedbydiceAstragaliAncientDiceThemostpopulardicegameoftheMiddleAges:wascalled“hazard”Arabic“alzhar”means“adie.”Romanicosahedraldie20facesWesterndiceareright-handed:ifthe1-spotisfaceupandthe2-spotisturnedtofacetheleftthenthe3-spotistotherightofit.Chinesediceareleft-handed:theywillhavethefacestheoppositewayround.RightandLeft-handedDiceTheProblemofthePointsChevalierdeMéréandBlaisePascalandPierredeFermat1654TwopeopleplayafairgameThefirsttowinsixpointstakesallthemoney.Howshouldthestakesbedividedifthegameisinterruptedwhenonehas5pointsandtheother3?HHH,HHT,HTH,TTT,THT,TTH,THH.HTTPlayerwith3pointshastowinallthenext3games.Hehas1/8chanceofdoingthat.Hisopponenthasa7/8chanceofwinning1moregame.Give7/8ofprizemoneytotheonewith5and1/8totheotherMoreChevalierdeMéréHewonlotsofmoneybettingonatleast1sixin4rollsofadiebasedpurelyonexperienceProbabilityofno6is5/6Probabilityofno6infourthrowsis5/65/65/65/6=(5/6)4=625/1296Probabilityofone6is1–625/1296=671/1296=0.5177>1/2Sohethoughtthatheshouldbetononeormoredouble6’soccurringin24rollsof2diceProbabilityofnodoublesixesin24throwsis(35/36)24=0.5086Probabilityofonedoublesixis1-(35/36)24=0.4914<1/2Afterawhilehestoppeddoingthis!WinningTheTossAustralianOpenJanuary2008PlayingFairWithaBiasedCoinUnequalprobabilityofHandT:p½

ProbabilityofHispProbabilityofTis1-pTosstwiceandignorepairsHHandTTProbabilityofHTisp(1-p)ProbabilityofTHis(1-p)pCallcombinationHT‘Newheads’CallcombinationTH‘Newtails’NewheadsandNewtailsareequallylikelyEfficiencyispoor(50%)–discardtheHHandTTsFakingRandomSequencesTHHTHTHTHTHTHTHTHTTTHTHTHTHTHTHHTHHTHTHTHHTHTHHHTTHHTHTTHHHTHTTTHTHHTHTTTHTHTHTHHTHTTTHHTHTHTHTTDotheselooklikerealrandomsequences?4.THHHHTTTTHTTHHHHTTHTHHTTHTTHTHHH5.HTTTTHHHTHTTHHHHTTTHTTTTHHTTTTTH6.TTHTTHHTHTTTTTHTTHHTTHTTTTTTTTHHSomeMoreCandidatesWith32tossesAretheyrandom?4.THHHHTTTTHTTHHHHTTHTHHTTHTTHTHHH5.HTTTTHHHTHTTHHHHTTTHTTTTHHTTTTTH6.TTHTTHHTHTTTTTHTTHHTTHHHHHHTTTTHSomeMoreCandidatesWith32tossesThechanceofarunofrheadsorrtailscomingupisjust½½½½….½,rtimes.Thisis1/2rIfwetossourcoinN>rtimesthereareNdifferentpossiblestartingpointsforarunofheadsortailsOurchanceofarunoflengthrisincreasedtoaboutN1/2r

ArunoflengthrisgoingtobecomelikelywhenN1/2risroughlyequalto1,thatiswhenN=2r.Notethat32=25Winning(andLosing)StreaksTheNasserHussainEffectEnglandcricketcaptainDuring2000-2001Athertontookoverforonegameafterhehadlost7andwonthetossNormalservicewasthenresumedThereisa1in214=16384chanceoflosingall14tossesButhecaptainedEngland101timesandthereisachanceofabout1in180ofalosingstreakof14“Flippinguseless,Nasser!”BBCCanYouAlwaysWin?Oravoideverlosing?TheWin-WinScenarioTheoddsfortherunnersarea1to1,a2to1,a3to1,andsoon,foranynumberofrunnersintherace.Iftheoddsare5to4thenweexpressthatasanaiof5/4to1Betafraction1/(ai+1)ofthetotalstakemoneyontherunnerwithoddsofaito1IfthereareNrunners,wewillalwaysmakeaprofitifQ=1/(a1+1)+1/(a2+1)+1/(a3+1)+….+1/(aN+1)<1

Winnings=(1/Q–1)ourtotalstakeExample:Fourrunnersandtheoddsforeachare6to1,7to2,2to1,and8to1and.Thenwehavea1=6,a2=7/2,a3=2anda4=8andQ=1/7+2/9+1/3+1/9=51/63<1Allocateourstakemoneywith1/7onrunner1,2/9onrunner2,1/3onrunner3,and1/9onrunner4Wewillwinatleast12/51ofthemoneywestaked(andofcoursewegetourstakemoneybackaswell).RaceFixing‘101’ThefavouriteisalwaysthelargestcontributortoQbecausea1isthesmallestoftheaisWecouldhaveQ>1withallrunnersincludedQ=1/(a1+1)+1/(a2+1)+…..>1Butifyouknowthefavouritehasbeenhobbledthenyou

calculateQexcludinga1whichcanresultinQfix=1/(a2+1)+1/(a3+1)+….<1Ifthereare4runnerswithodds3to1,7to1,3to2,and1to1Q=1/4+1/8+2/5+1/2=51/40>1Sowecan’tguaranteeawinningreturnDopethefavouriteandplaceyoumoneyontheotherthreerunnersonly,betting1/4ofourstakemoneyonrunner1,1/8onrunner2,and2/5onrunner3Youarereallybettingona3-horseracewithQfix=1/4+1/8+2/5=31/40<1Whatevertheoutcomeyouwillneverdoworsethanwinningyourstakemoneyplus

{(40/31)-1}Stakemoney=9/31Stakemoney

OutcomeBookmaker1’soddsBookmaker2’soddsOxfordwin1.251.43Cambridgewin3.92.85QofBookie11.056>1Hegains5.6%QofBookie2Hegains5.1%1.051>1AMixedStrategyBackOxfordwithBk2andCambridgewithBk1

Q=1.43-1+3.9-1

Q

=0.956<1Youcanearn4.6%Bet100onOxfordwithbookie2and100x1.43/3.9=36.67onCambridgeatbookie1.

IfOxfordwin,youcollect100x1.43=143frombookie2.IfCambridgewin,youcouldcollect36.67x3.9=143frombookie1.Youinvested136.67andcollect143,aprofitof6.33(4.6%)nomatterwhattheoutcome.WhenBookiesDisagreeWhatAbouttheQ>1SituationsThisisthemoney-launderingcaseYouareguaranteedalossof(1-1/Q)ofyourstakemoneyThatisthecostofthelaunderingandcarriesnoriskofgreaterlossWeirdJudgingMeansIceSkatingLadiesFigureSkatingSaltLakeCityOlympicsSkaterShortLongTotalKwan0.52.02.5Hughes2.01.03.0Cohen1.53.04.5Slutskaya1.0??Beforethelastcompetitorskates…LowestscoresleadSkaterShortLongTotalHughes2.01.03.0Slutskaya1.02.03.0Kwan0.53.03.5Cohen1.54.05.5AndafterSlutskayaskates…Hugheswinsbytie-break!Slut